Final answer:
The question asks for the cardinality of the multiplicative group (Z × 2027, ·) generated by 314 mod 2027, which involves finding the order of the element 314 in the group modulo 2027.
Step-by-step explanation:
The question pertains to the concept of group theory in abstract algebra, specifically regarding the cardinality of a group. The group in question, denoted as (Z × 2027, ·), is the multiplicative group modulo 2027 generated by g = 314 mod 2027. To find the cardinality of this group, we need to determine the order of the element 314 in the group. This entails finding the smallest positive integer n such that g^n ≡ 1 (mod 2027). Here, n would represent the cardinality, which is the number of elements in the cyclic subgroup generated by g.
We must check successive powers of 314 modulo 2027 until we reach 1, starting with 314^1, 314^2, and so on. However, since 2027 is a specific number and the calculation may require advanced computational tools or algorithms, the exact value of the cardinality is not provided here, as it involves computations beyond the scope of a traditional explanation.